Deck 11: General Rules of Probability

Question 1

Spelling mistakes in a text are either "nonword errors" or "word errors." A nonword error produces a string of letters that is not a word, such as "the" typed as "teh." Word errors produce the wrong word, such as "loose" typed as "lose." Nonword errors make up 25% of all errors. A human proofreader will catch 80% of nonword errors and 50% of word errors. Of all the errors that the proofreader catches, what percent are word errors?

A)34.8%
B)37.5%
C)57.5%
D)65.2%

Answer 1

65.2%

Question 2

A roulette wheel has 38 slots in which the ball can land. Two of the slots are green, 18 are red, and 18 are black. The ball is equally likely to land in any slot. The roulette wheel is going to be spun twice, and the outcomes of the two spins are independent. The probability that the ball lands on black the first time and on green the second time is:

A)0.0249.
B)0.2244.
C)0.277.
D)0.5263.

Answer 2

0.0249.

Question 3

A group of college DJs surveyed students to find out what music to plan for their upcoming parties. Thirty percent of the students preferred dubstep, 25% of the students liked trance music, and 20% wanted to hear only house music. Fifteen percent of the respondents selected both dubstep and trance. The proportion of students that like neither trance music nor dubstep is:

A)0%.
B)20%.
C)40%.
D)60%.

Answer 3

60%.

Question 4

A group of college DJs surveyed students to find out what music to plan for their upcoming parties. Thirty percent of the students preferred dubstep, 25% of the students liked trance music, and 20% wanted to hear only house music. Fifteen percent of the respondents selected both dubstep and trance. The conditional probability that a student likes dubstep, given that he or she likes trance music, is:

A)55%.
B)60%.
C)65%.
D)70%.

Question 5

A group of freshmen at a local university consider joining the equestrian team. Thirty-five percent of students choose Western riding, 45% choose dressage, and 40% choose jumping. Fifteen percent choose both dressage and jumping, while 10% choose Western and dressage. No one chooses Western and jumping. There are no horses suitable for two styles, and each student is assigned to one horse. If five students decide to join the team, what is the probability that at least one student joins the dressage team?

A)0.0185
B)0.0503
C)0.0412
D)0.9497

Question 6

A system has two components that operate in parallel, as shown in the following diagram. Because the components operate in parallel, at least one of the components must function properly if the system is to function properly. The probability of failure for components 1 and 2, during one period of operation, is 0.2 and 0.03, respectively. Let F1 denote the event that component 1 fails during one period of operation and F2 denote the event that component 2 fails during one period of operation. The component failures are independent. The probability that the system fails during one period of operation is closest to:

A)0.23.
B)0.224.
C)0.06.
D)0.006.

Question 7

A group of freshmen at a local university consider joining the equestrian team. Thirty-five percent of students choose Western riding, 45% choose dressage, and 40% choose jumping. Fifteen percent choose both dressage and jumping, while 10% choose Western and dressage. No one chooses Western and jumping. There are no horses suitable for two styles, and each student is assigned to one horse. If two students decide to join the team, what is the probability that one student joins the dressage team and the other student does not?

A)0.2275
B)0.1925
C)0.2475
D)0.1525

Question 8

A group of freshmen at a local university consider joining the equestrian team. Thirty-five percent of students choose Western riding, 45% choose dressage, and 50% choose jumping. Twenty percent choose both dressage and jumping, while 10% choose Western and dressage. No one chooses Western and jumping. There are no horses suitable for two styles, and each student is assigned to one horse. If two students decide to join the team, what is the probability that both are Western and dressage riders, if they independently decide?

A)0.1
B)0.2
C)0.01
D)0.8

Question 9

A group of college DJs surveyed students to find out what music to plan for their upcoming parties. Thirty percent of the students preferred dubstep, 25% of the students liked trance music, and 20% wanted to hear only house music. Fifteen percent of the respondents selected both dubstep and trance. The proportion of students that didn't select any of the music options available on the survey was:

A)0%.
B)20%.
C)40%.
D)60%.

Question 10

The following table gives the sex and age group of college students at a Midwestern university. A student is to be selected at random. The probability that the selected student is 25 to 34 years old is:

A)0.21.
B)0.25.
C)0.56.
D)0.623.

Question 11

A system has two components that operate in parallel, as shown in the following diagram. Because the components operate in parallel, at least one of the components must function properly if the system is to function properly. The probability of failure for components 1 and 2, during one period of operation, is 0.2 and 0.03, respectively. Let F1 denote the event that component 1 fails during one period of operation and F2 denote the event that component 2 fails during one period of operation. The component failures are independent. The probability that the system functions properly during one period of operation is closest to:

A)0.994.
B)0.97.
C)0.94.
D)0.776.

Question 12

A group of freshmen at a local university consider joining the equestrian team. Thirty-five percent of students choose Western riding, 45% choose dressage, and 50% choose jumping. Twenty percent choose both dressage and jumping, while 10% choose Western and dressage. No one chooses Western and jumping. There are no horses suitable for two styles, and each student is assigned to one horse. What is the probability that a student chooses dressage or jumping?

A)0.95
B)0.75
C)0.23
D)0.9

Question 13

Spelling mistakes in a text are either "nonword errors" or "word errors." A nonword error produces a string of letters that is not a word, such as "the" typed as "teh." Word errors produce the wrong word, such as "loose" typed as "lose." Nonword errors make up 25% of all errors. A human proofreader will catch 80% of nonword errors and 50% of word errors. What percent of errors will the proofreader catch?

A)20%
B)37.5%
C)57.5%
D)80%

Question 14

A system has two components that operate in parallel, as shown in the following diagram. Because the components operate in parallel, at least one of the components must function properly if the system is to function properly. The probability of failure for components 1 and 2, during one period of operation, is 0.2 and 0.03, respectively. Let F1 denote the event that component 1 fails during one period of operation and F2 denote the event that component 2 fails during one period of operation. The component failures are independent. The event corresponding to the above system failing during one period of operation is:

A)F1 and F2.
B)F1 or F2.
C)not F1 or not F2.
D)not F1 and not F2.

Question 15

A group of college DJs surveyed students to find out what music to plan for their upcoming parties. Thirty percent of the students preferred dubstep, 25% of the students liked trance music, and 20% wanted to hear only house music. Fifteen percent of the respondents selected both dubstep and trance. The proportion of students that like either dubstep or trance is:

A)40%.
B)55%.
C)70%.
D)100%.

Question 16

A group of freshmen at a local university consider joining the equestrian team. Thirty-five percent of students choose Western riding, 45% choose dressage, and 50% choose jumping. Twenty percent choose both dressage and jumping, while 10% choose Western and dressage. No one chooses Western and jumping. There are no horses suitable for two styles, and each student is assigned to one horse. What is the probability that a student chooses neither dressage nor Western riding?

A)0.3
B)0.5
C)0.2
D)0.05

Question 17

A group of freshmen at a local university consider joining the equestrian team. Thirty-five percent of students choose Western riding, 45% choose dressage, and 40% choose jumping. Fifteen percent choose both dressage and jumping, while 15% choose Western and dressage. No one chooses Western and jumping. There are no horses suitable for two styles, and each student is assigned to one horse. If two students decide to join the team, what is the probability that one student joins jumping only and the other student joins the Western team, if they independently decide?

A)0.175
B)0.125
C)0.0375
D)0.105

Question 18

A roulette wheel has 38 slots in which the ball can land. Two of the slots are green, 18 are red, and 18 are black. The ball is equally likely to land in any slot. The roulette wheel is going to be spun twice, and the outcomes of the two spins are independent. The probability that the ball lands on red at least once is:

A)0.4986.
B)0.723.
C)0.7756.
D)0.9474.

Question 19

A group of college DJs surveyed students to find out what music to plan for their upcoming parties. Thirty percent of the students preferred dubstep, 25% of the students liked trance music, and 20% wanted to hear only house music. Fifteen percent of the respondents selected both dubstep and trance. The proportion of students that like trance music but not dubstep is:

A)5%.
B)10%.
C)15%.
D)20%.

Question 20

A system has two components that operate in parallel, as shown in the following diagram. Because the components operate in parallel, at least one of the components must function properly if the system is to function properly. The probability of failure for components 1 and 2, during one period of operation, is 0.2 and 0.03, respectively. Let F1 denote the event that component 1 fails during one period of operation and F2 denote the event that component 2 fails during one period of operation. The component failures are independent. The event corresponding to the above system functioning properly during one period of operation is:

A)F1 and F2.
B)F1 or F2.
C)not F1 or not F2.
D)not F1 and not F2.

Question 21

An event A will occur with probability 0.5. An event B will occur with probability 0.6. The probability that both A and B will occur is 0.1. The conditional probability of B, given A, is:

A)5/6.
B)1/5.
C)1/6.
D)The answer cannot be determined from the information given.

Question 22

The following table gives the sex and age group of college students at a Midwestern university. A student is to be selected at random. All of the outcomes counted in this table are:

A)discrete.
B)disjoint.
C)finite.
D)None of the answer options is correct.

Question 23

Veterinarians suspect that enteroliths (calcifications in the gut of horses) are related to diet-in particular, alfalfa. To investigate this suspicion, a group of veterinarians collected information on the diet of horses and whether the horses developed enteroliths. The table below displays the findings (a case is a horse with enteroliths, a control is without enteroliths). The probability that a randomly selected horse gets more than 50% alfalfa is:

A)0.33.
B)0.5.
C)0.8.
D)0.23.

Question 24

In a particular game, a six-sided fair die is tossed. If the number of spots showing is six, you win $6; if the number of spots showing is five, you win $3; if the number of spots showing is four, you win $2; and if the number of spots showing is three, you win $1. If the number of spots showing is one or two, you win nothing. You are going to play the game twice. The probability that you win at least $9 in total on the two plays of the game is:

A)1/6.
B)1/3.
C)1/36.
D)1/12.

Question 25

The following table gives the sex and age group of college students at a Midwestern university. A student is to be selected at random. Given that the selected student is female, the conditional probability that she is 25 to 34 years old is:

A)0.545.
B)0.204.
C)0.114.
D)0.008.

Question 26

Veterinarians suspect that enteroliths (calcifications in the gut of horses) are related to diet-in particular, alfalfa. To investigate this suspicion, a group of veterinarians collected information on the diet of horses and whether the horses developed enteroliths. The table below displays the findings (a case is a horse with enteroliths, a control is without enteroliths). The probability that a randomly selected horse has enteroliths, given that it is fed more than 50% alfalfa, is:

A)0.23.
B)0.348.
C)0.08.
D)0.2.

Question 27

In the Virginia instant lottery, there are 10 different $5 Scratcher games. Your favorite, "Hit the Jackpot," is advertised to have a 1-in-4.37 chance of winning, and a 1-in-664,457 chance of hitting the top prize of $200,000. All of the seven top prizes are still available. If you buy five of these tickets and outcomes are independent, the probability that you lose all five times is:

A)0.229.
B)0.273.
C)0.459.
D)0.771.

Question 28

A stack of four cards contains two red cards and two black cards. I select two cards, one at a time, and do not replace the first card selected before selecting the second card. Consider the events: A = the first card selected is red.
B = the second card selected is red.
The events A and B are:

A)independent.
B)disjoint.
C)conditionals.
D)None of the answer options is correct.

Question 29

In the Virginia instant lottery, there are 10 different $5 Scratcher games. Your favorite, "Hit the Jackpot," is advertised to have a 1-in-4.37 chance of winning and a 1-in-664,457 chance of hitting the top prize of $200,000. All of the seven top prizes are still available. If you buy five of these tickets and outcomes are independent, the probability of winning at least once in these five draws is:

A)0.353.
B)0.459.
C)0.727.
D)0.771.

Question 30

In a particular game, a six-sided fair die is tossed. If the number of spots showing is six, you win $6; if the number of spots showing is five, you win $3; if the number of spots showing is four, you win $2; and if the number of spots showing is three, you win $1. If the number of spots showing is one or two, you win nothing. You are going to play the game twice. The probability that you win something on each of the two plays of the game is:

A)1/6.
B)1/3.
C)4/9.
D)1/4.

Question 31

Event A occurs with probability 0.2. Event B occurs with probability 0.9. Events A and B:

A)are disjoint.
B)cannot be independent.
C)cannot be disjoint.
D)are reciprocating.

Question 32

Veterinarians suspect that enteroliths (calcifications in the gut of horses) are related to diet-in particular, alfalfa. To investigate this suspicion, a group of veterinarians collected information on the diet of horses and whether the horses developed enteroliths. The table below displays the findings (a case is a horse with enteroliths, a control is without enteroliths). The probability that a randomly selected horse is fed more than 50% alfalfa, given that it has enteroliths, is:

A)0.2.
B)0.23.
C)0.08.
D)0.4.

Question 33

The following table gives the sex and age group of college students at a Midwestern university. A student is to be selected at random. The probability that the selected student is a female who is 15 to 17 years old is:

A)0.005.
B)0.01.
C)0.56.
D)0.593.

Question 34

After observing the defects within individual candies in many bags of M&Ms, it has been determined that 11% of all candies are defective, that the probability of observing an M&M with a missing letter is 22%, and that the probability of observing a cracked M&M, given that you already know it is defective, is 70%. You can calculate the probability that you randomly select an M&M that is cracked because:

A)the events are discrete.
B)the events are independent.
C)the outcomes are disjoint.
D)None of the answer options is correct.

Question 35

Students at a local university have the option of taking freshman seminars during their first year in college. A survey of the freshmen revealed the following: Among the social science majors, 50% chose to take a freshman seminar; among the humanities majors, 65% chose to take a freshman seminar; and among the physical science majors, it was 30%. Freshmen make up 32% of undergraduates. The probability of taking a freshman seminar, if a student is a science major, is:

A)a total probability.
B)a complete probability.
C)a joint probability.
D)a conditional probability.

Question 36

Suppose we toss a coin and roll a die. Let A be the event that the number of spots showing on the die is three or less, and let B be the event that the coin comes up heads. The events A and B are:

A)disjoint.
B)conditional.
C)independent.
D)reciprocals.

Question 37

Rolling a fair six-sided die three times is an example of:

A)three events that are independent.
B)three outcomes that are disjoint.
C)three events that are independent, with three outcomes that are disjoint.
D)None of the answer options is correct.

Question 38

In the Virginia instant lottery, there are 10 different $5 Scratcher games. Your favorite, "Hit the Jackpot," is advertised to have a 1-in-4.37 chance of winning, and a 1-in-664,457 chance of hitting the top prize of $200,000. All of the seven top prizes are still available. If you buy five of these tickets and outcomes are independent, the probability that you win all five times is:

A)6.27%.
B)0.627%.
C)0.0627%.
D)0.0063%.

Question 39

An event A will occur with probability 0.5. An event B will occur with probability 0.6. The probability that both A and B will occur is 0.1. We may conclude that:

A)events A and B are independent.
B)events A and B are disjoint.
C)either A or B always occurs.
D)None of the answer options is correct.

Question 40

Veterinarians suspect that enteroliths (calcifications in the gut of horses) are related to diet-in particular, alfalfa. To investigate this suspicion, a group of veterinarians collected information on the diet of horses and whether the horses developed enteroliths. The table below displays the findings (a case is a horse with enteroliths, a control is without enteroliths). There is a relationship between being fed alfalfa and having enteroliths because:

A)P(enteroliths | > 50% alfalfa)> P(enteroliths | 25%-50% alfalfa)> P(enteroliths | < 25% alfalfa).
B)fewer horses get more than 50% alfalfa.
C)the proportion of horses that have enteroliths and eat more than 50% alfalfa is the largest probability in the "Case" row of the table.
D)None of the answer options is correct.

Question 41

University degree requirements typically are different for Bachelor of Science degrees and Bachelor of Arts degrees. Some students get a Bachelor of Arts and Science degree, which requires meeting graduation criteria for both degrees. A student advisor needs to know the probability a newly admitted student is interested in such a program, so that the student can be properly advised. A study of previous years finds that the probability a student gets a Bachelor of Science degree is P(Science) = 0.3 and the probability a student gets a Bachelor of Arts degree is P(Arts) = 0.6. The probability of getting a Bachelor of Science degree, given that a student is getting a Bachelor of Arts degree, is given by P(Science | Arts) = 0.12. The probability of a student getting both a Bachelor of Arts degree and a Bachelor of Science degree is:

A)0.12.
B)0.6.
C)0.072.
D)0.3.

Question 42

University degree requirements typically are different for Bachelor of Science degrees and Bachelor of Arts degrees. Some students get a Bachelor of Arts and Science degree, which requires meeting graduation criteria for both degrees. A student advisor needs to know the probability a newly admitted student is interested in such a program, so that the student can be properly advised. A study of previous years finds that the probability a student gets a Bachelor of Science degree is P(Science) = 0.3 and the probability a student gets a Bachelor of Arts degree is P(Arts) = 0.6. The study also shows that the probability a student gets no degree is P(no) = 0.2. The probability a student gets a Bachelor of Arts and Science degree is:

A)0.2.
B)0.3.
C)0.1.
D)0.6.

Question 43

University degree requirements typically are different for Bachelor of Science degrees and Bachelor of Arts degrees. Some students get a Bachelor of Arts and Science degree, which require meeting graduation criteria for both degrees. A student advisor needs to know the probability a newly admitted student is interested in such a program, so that the student can be properly advised. A study of previous years finds that the probability a student gets a Bachelor of Science degree is P(Science) = 0.3 and the probability a student gets a Bachelor of Arts degree is P(Arts) = 0.6. The study also shows that the probability a student gets no degree is P(no) = 0.2. Some probability calculations show the probability of getting a Bachelor of Arts and Sciences to be P(Arts & Science) = 0.1. Getting a Bachelor of Arts degree and getting a Bachelor of Science degree are:

A)disjoint events.
B)independent events.
C)adverse events.
D)dependent events.

Question 44

Let A be the event that a flight from New York to San Francisco arrives on time and let B be the event that it is a clear day in San Francisco. Suppose the probability of a clear day is P(B) = 0.6 and the probability a plane arrives on time is P(A) = 0.7. We also know that the probability a plane arrives on time on a cloudy day is P(A | B^C) = 0.5. The probability a plane arrives on time and it is a cloudy day is:

A)0.6.
B)0.5.
C)0.2.
D)0.42.

Question 45

University degree requirements typically are different for Bachelor of Science degrees and Bachelor of Arts degrees. Some students get a Bachelor of Arts and Science degree, which requires meeting graduation criteria for both degrees. A student advisor needs to know the probability a newly admitted student is interested in such a program, so that the student can be properly advised. A study of previous years finds that the probability a student gets a Bachelor of Science degree is P(Science) = 0.3 and the probability a student gets a Bachelor of Arts degree is P(Arts) = 0.6. The study also shows that the probability a student gets no degree is P(no) = 0.2. The probability a student gets a Bachelor of Arts degree or a Bachelor of Science degree is:

A)0.9.
B)0.8.
C)0.3.
D)0.6.

Question 46

In the Virginia instant lottery, there are 10 different $5 Scratcher games. Your favorite, "Hit the Jackpot," is advertised to have a 1-in-4.37 chance of winning, and a 1-in-664,457 chance of hitting the top prize of $200,000. All of the seven top prizes are still available. If you buy five of these tickets and outcomes are independent, the probability that you will not win the jackpot is:

A)0.9.
B)0.95.
C)0.99.
D)0.999.

Question 47

University degree requirements typically are different for Bachelor of Science degrees and Bachelor of Arts degrees. Some students get a Bachelor of Arts and Science degree, which requires meeting graduation criteria for both degrees. A student advisor needs to know the probability a newly admitted student is interested in such a program, so that the student can be properly advised. A study of previous years finds that the probability a student gets a Bachelor of Science degree is P(Science) = 0.30 and the probability a student gets a Bachelor of Arts degree is P(Arts) = 0.60. If getting a Bachelor of Science degree is independent of getting a Bachelor of Arts degree, the probability of getting a Bachelor of Arts and Science degree is:

A)0.1.
B)0.18.
C)0.2.
D)None of the answer options is correct.

Question 48

University degree requirements typically are different for Bachelor of Science degrees and Bachelor of Arts degrees. Some students get a Bachelor of Arts and Science degree, which requires meeting graduation criteria for both degrees. A student advisor needs to know the probability a newly admitted student is interested in such a program, so that the student can be properly advised. A study of previous years finds that the probability a student gets a Bachelor of Science degree is P(Science) = 0.3 and the probability a student gets a Bachelor of Arts degree is P(Arts) = 0.6. The study also shows that the probability a student gets no degree is P(no) = 0.2. Some probability calculations show the probability of getting a Bachelor of Arts and Science degree to be P(Arts & Science) = 0.1. Given this result, we conclude that:

A)getting a Bachelor of Science is independent of getting a Bachelor of Arts.
B)getting a Bachelor of Arts and getting a Bachelor of Science are mutually exclusive.
C)getting a Bachelor of Arts and getting a Bachelor of Science are disjoint.
D)None of the answer options is correct.

Question 49

University degree requirements typically are different for Bachelor of Science degrees and Bachelor of Arts degrees. Some students get a Bachelor of Arts and Science degree, which requires meeting graduation criteria for both degrees. A student advisor needs to know the probability a newly admitted student is interested in such a program, so that the student can be properly advised. A study of previous years finds that the probability a student gets a Bachelor of Science degree is P(Science) = 0.3 and the probability a student gets a Bachelor of Arts degree is P(Arts) = 0.6. The probability of getting a Bachelor of Science degree, given that a student is getting a Bachelor of Arts degree, is given by P(Science | Arts) = 0.3. The probability that a student will get no degree is:

A)0.1.
B)0.28.
C)0.3.
D)0.4.

Question 50

University degree requirements typically are different for Bachelor of Science degrees and Bachelor of Arts degrees. Some students get a Bachelor of Arts and Science degree, which requires meeting graduation criteria for both degrees. A student advisor needs to know the probability a newly admitted student is interested in such a program, so that the student can be properly advised. A study of previous years finds that the probability a student gets a Bachelor of Science degree is P(Science) = 0.3 and the probability a student gets a Bachelor of Arts degree is P(Arts) = 0.6. The study also shows that the probability a student gets no degree is P(no) = 0.2. Some probability calculations show the probability of getting a Bachelor of Arts and Science degree to be P(Arts & Science) = 0.1. The probability of getting a Bachelor of Arts and Science degree, given that a student is getting a Bachelor of Science degree, is:

A)0.333.
B)0.167.
C)0.111.
D)0.125.

Question 51

University degree requirements typically are different for Bachelor of Science degrees and Bachelor of Arts degrees. Some students get a Bachelor of Arts and Science degree, which requires meeting graduation criteria for both degrees. A student advisor needs to know the probability a newly admitted student is interested in such a program, so that the student can be properly advised. A study of previous years finds that the probability a student gets a Bachelor of Science degree is P(Science) = 0.3 and the probability a student gets a Bachelor of Arts degree is P(Arts) = 0.6. The study also shows that the probability a student gets no degree is P(no) = 0.2. The probability a student gets a Bachelor of Arts and Science degree is:

A)0.2.
B)0.5.
C)0.3.
D)0.1.

Question 52

University degree requirements typically are different for Bachelor of Science degrees and Bachelor of Arts degrees. Some students get a Bachelor of Arts and Science degree, which requires meeting graduation criteria for both degrees. A student advisor needs to know the probability a newly admitted student is interested in such a program, so that the student can be properly advised. A study of previous years finds that the probability a student gets a Bachelor of Science degree is P(Science) = 0.3 and the probability a student gets a Bachelor of Arts degree is P(Arts) = 0.6. The study also shows that the probability a student gets no degree is P(no) = 0.2. The probability a student gets only a Bachelor of Arts degree is:

A)0.5.
B)0.6.
C)0.3.
D)0.1.